Solve x^2 + x 2 + 1 = 0 - Vidyadip

Question

Solve ${x^2} + \frac{x}{{\sqrt 2 }} + 1 = 0$

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Answer

Here ${x^2} + \frac{x}{{\sqrt 2 }} + 1 = 0$
Comparing the given quadratic equation with $ax^2 + bx + c = 0$ we have
$a = 1,b = \frac{1}{{\sqrt 2 }}$ and c = 1
$\therefore x=\frac{\frac{-1}{\sqrt2}+\sqrt{\left(\frac1{\sqrt2}\right)^2-4\times1\times1}}{2\times1}$
$ = \frac{{\frac{{ - 1}}{{\sqrt 2 }} \pm \sqrt {\frac{1}{2} - 4} }}{2}$
$ = \frac{{\frac{{ - 1}}{{\sqrt 2 }} \pm \sqrt {\frac{{ - 7}}{2}} }}{2} = \frac{{ - 1 \pm \sqrt 7 i}}{{2\sqrt 2 }}$
Thus $x = \frac{{ - 1 + \sqrt 7 i}}{{2\sqrt 2 }}$ and $x = \frac{{ - 1 - \sqrt 7 i}}{{2\sqrt 2 }}$

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